- Last week, a very big number — over 23 million digits long — became the “largest known prime number”.
- The number, 277,232,917-1, was discovered using a software called GIMPS, which allows volunteers to search for Mersenne prime numbers (more on that below). Jonathan Pace, a volunteer from Tennessee, made the discovery on December 26, and it was further confirmed using four different programs on four different pieces of hardware.
- In case you want to look at the 23-million-digit number, here is the link: http://www.mersenne.org/primes/digits/M77232917.zip
What are prime numbers and why are they important?
- A prime number is a number that can only be divided by itself and by 1. For example: 2, 3, 5, 7, 11, and so on.
- British mathematician Marcus du Sautoy, in his book The Music of the Primes,writes, “Prime numbers are the very atoms of arithmetic. The prime numbers 2, 3 and 5 are the hydrogen, helium and lithium in the mathematician’s laboratory. Mastering these building blocks offers the…hope of discovering new ways…through the vast complexities of the mathematical world.”
- Dr. Baskar Balasubramanyam, Assistant Professor at the Department of Mathematics, IISER Pune, explained in detail about the new discovery in an e-mail to The Hindu:
Why is the new number called a Mersenne prime number?
- Mersenne prime is a prime number of the form 2n-1. For example, 7 = 23-1 and is a prime, so it is a Mersenne prime.
- One the other hand 11 is a prime, but it is not of the form 2n-1. So it is not a Mersenne prime. Not all numbers of the form 2n-1 are primes either. For example, 24-1 = 15 is not a prime.
- The GIMPS project looks at such numbers to figure out which of them are going to be primes.
So, through this software can we find bigger prime numbers?
- One of the oldest theorems in mathematics (the Euclid theorem) says that there are infinitely many primes. So we are going to find larger and larger primes.
- For number theorists, it is also important to understand if there are infinitely many primes that fit a particular pattern. For example, are there infinitely many primes of the form 4n+1? The answer is yes.
- We still don’t know if there are infinitely many Mersenne primes. Another ‘family’ that is of much interest are the Twin Primes (primes that are separated by 2 like 11 and 13).
Can you tell me about the applications of prime numbers?
- One of the major applications of primality testing (testing whether a number is prime) is in cryptography (Cryptography, which is derived from the Greek word for the study of secret messaging, involves sharing information via secret codes).
- This is based on the following principle: multiplying two numbers is easy, factoring a number is hard. For cryptographic applications, we need a number N that is a product of two primes p and q (N = pq). The value of N is public information, but it is very difficult to find p and q just by knowing the value of N — there are lots of possibilities for p and q.
- Our credit cards, cell phones, all depend on cryptography.